Complete convergence in the strong law of large numbers for double sums indexed by a sector with functional boundaries. (Ukrainian, English) Zbl 1050.60034

Indlekofer, K.-H.; Klesov, O. I.

Teor. Jmovirn. Mat. Stat. 68, 44-48 (2003); translation in Theory Probab. Math. Stat. 68, 49-54 (2004).

Let \(\{X(i,j);\;i\geq1, j\geq1\}\) be i.i.d. random variables; \(S(m,n)= \sum_{i=1}^{m} \sum_{j=1}^{n}X(i,j)\); \(f: R\to R\) be a function such that \(f(x)\geq x\) for all \(x\geq1\); \(A=\{(m,n): m\leq n\leq f(m)\}\); \(\tau_{k}=\text{card}\{(m,n): ij=k, (i,j)\in A\}\). The sequence of random variables \(\{U(m,n)\); \(m\geq1\), \(n\geq1\}\) is called completely convergent to 0 on the set \(A\) if \(\sum_{(m,n)\in A}\text{P}(| U(m,n)| \geq\varepsilon)<\infty\) for all \(\varepsilon>0\). The main result of this paper is the following: The complete convergence of \(S(m,n)/mn\) to 0 on the set \(A\) is equivalent to the conditions \(EX(i,j)=0\), \(\sum_{k=1}^{\infty}k\tau_{k}\text{P} (| X(i,j)|\geq\delta)<\infty\) for all \(\delta>0\).

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60F15	Strong limit theorems
60K05	Renewal theory

Keywords:

complete convergence; strong law of large numbers; double sums indexed by a sector with functional boundaries